On some conjectures on the monotonicity of some arithmetical sequences (Q2869143)

This paper establishes several conjectures due to Zhi-Wei Sun regarding the monotonicity of several sequences of the form \(a_{n+1}^{1/(n+1)}/a_n^{1/n}\) or \(a_n^{1/n}\).NEWLINENEWLINESpecifically, if \(B_n\) denotes the Bernoulli numbers, \(T_n\) the tangent numbers and \(E_n\) the Euler numbers, then the sequences \(|B_{2n}|^{1/n}\), \(|T_{2n-1}|^{1/n}\) and \(|E_{2n}|^{1/n}\) are all increasing, whereas the sequences \(|B_{2n+2}|^{1/(n+1)}/|B_{2n}|^{1/n}\), \(|T_{2n+1}|^{1/(n+1)}/|T_{2n-1}|^{1/n}\) and \(|E_{2n+2}|^{1/(n+1)}/|E_{2n}|^{1/n}\) are decreasing.NEWLINENEWLINEThis is shown by obtaining precise asymptotic estimates for the Bernoulli, tangent and Euler numbers.NEWLINENEWLINEAnalogous statements are obtained for several further sequences (partly starting from a certain index) including the Apéry, Delannoy, Franel, Motzkin and Schröder numbers as well as the trinomial coefficients.